PD-sets for codes related to flag-transitive symmetric designs Nina - PowerPoint PPT Presentation

Introduction Codes from graphs Flag-transitive symmetric designs Examples PD-sets for codes related to flag-transitive symmetric designs Nina Mostarac (nmavrovic@math.uniri.hr) Dean Crnkovi´ c (deanc@math.uniri.hr) Department of Mathematics, University of Rijeka, Croatia Supported by CSF (Croatian Science Foundation), Grant 6732 Finite Geometry & Friends A Brussels summer school on finite geometry June 18, 2019 Nina Mostarac (nmavrovic@math.uniri.hr) 1/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Introduction • permutation decoding was introduced in 1964 by MacWilliams • it uses sets of code automorphisms called PD-sets • the problem of existence of PD-sets and finding them • we will prove the existence of PD-sets for all codes generated by the incidence matrix of an incidence graph of a flag-transitive symmetric design and construct some examples Nina Mostarac (nmavrovic@math.uniri.hr) 2/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Refrences D. Crnkovi´ [1] c, N. Mostarac, PD-sets for codes related to flag-transitive symmetric designs, Trans. Comb. , 7 (2018) 37–50. [2] P . Dankelmann, J.D. Key and B.G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Cryptogr. , 68 (2013) 373–393. • for prime p let C p ( G ) be the p -ary code spanned by the rows of the incidence matrix G of a graph Γ • we will show that if Γ is the incidence graph of a flag-transitive symmetric design D , then any flag-transitive automorphism group of D can be used as a PD-set for full error correction for the linear code C p ( G ) (with any information set) Nina Mostarac (nmavrovic@math.uniri.hr) 3/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Codes Definition 1 Let p be a prime. A p -ary linear code C of length n and dimension k is a k -dimensional subspace of the vector space ( F p ) n . Definition 2 • Let x = ( x 1 , ..., x n ) and y = ( y 1 , ..., y n ) ∈ F n p . The Hamming distance between words x and y is the number d ( x , y ) = |{ i : x i � = y i }| . • The minimum distance of the code C is defined by d = min { d ( x , y ) : x , y ∈ C , x � = y } . • Notation: [ n , k , d ] p code • it can detect at most d − 1 errors in one codeword and correct at � d − 1 � most t = errors 2 Nina Mostarac (nmavrovic@math.uniri.hr) 4/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Graphs We will discuss undirected graphs, with no loops and multiple edges. Definition 3 Edge connectivity λ (Γ) of a connected graph Γ is the minimum number of edges that need to be removed to disconnect the graph. Remark 1 For every graph Γ : λ (Γ) ≤ δ (Γ) . Nina Mostarac (nmavrovic@math.uniri.hr) 5/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Codes from incidence matrices of graphs Let G be the incidence matrix of a graph Γ = ( V , E ) over F p , p prime and the code C p ( G ) the row-span of G over F p . Theorem 2.1 (Dankelmann, Key, Rodrigues [2](Result 1)) Let Γ = ( V , E ) be a connected graph and G its incidence matrix. Then: 1 dim ( C 2 ( G )) = | V | − 1 ; 2 for odd p, dim ( C p ( G )) = | V | if Γ is not bipartite, and dim ( C p ( G )) = | V | − 1 if Γ is bipartite. Nina Mostarac (nmavrovic@math.uniri.hr) 6/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Codes from incidence matrices of graphs Theorem 2.2 (Dankelmann, Key, Rodrigues [2](Theorem 1)) Let Γ = ( V , E ) be a connected graph, G a | V | × | E | incidence matrix for G. Then: 1 C 2 ( G ) is a [ | E | , | V | − 1 , λ (Γ)] 2 code; 2 if Γ is super- λ , then C 2 ( G ) is a [ | E | , | V | − 1 , δ (Γ)] 2 code, and the minimum words are the rows of G of weight δ (Γ) . Nina Mostarac (nmavrovic@math.uniri.hr) 7/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Codes from incidence matrices of graphs Theorem 2.3 (Dankelmann, Key, Rodrigues [2](Theorem 2)) Let Γ = ( V , E ) be a connected bipartite graph, G a | V | × | E | incidence matrix for G, and p an odd prime. Then: 1 C p ( G ) is a [ | E | , | V | − 1 , λ (Γ)] p code; 2 if Γ is super- λ , then C p ( G ) is a [ | E | , | V | − 1 , δ (Γ)] p code, and the minimum words are the non-zero scalar multiples of the rows of G of weight δ (Γ) . Nina Mostarac (nmavrovic@math.uniri.hr) 8/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Codes from incidence matrices of graphs Theorem 2.4 (Dankelmann, Key, Rodrigues [2](Result 3)) Let Γ = ( V , E ) be a connected bipartite graph. Then λ (Γ) = δ (Γ) if one of the following conditions holds: 1 V consists of at most two orbits under Aut (Γ) , and in particular if Γ is vertex-transitive; 2 every two vertices in one of the two partite sets of Γ have a common neighbour; 3 diam (Γ) ≤ 3 ; 4 Γ is k-regular and k ≥ n + 1 ; 4 5 Γ has girth g and diam (Γ) ≤ g − 1 . Nina Mostarac (nmavrovic@math.uniri.hr) 9/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Information sets Definition 4 Let C ⊆ F n p be a linear [ n , k , d ] code. For I ⊆ { 1 , ..., n } let p → F | I | p I : F n p , x �→ x | I , be an I -projection of F n p . Then I is called an information set for C if | I | = k and p I ( C ) = F | I | p . The set of the first k coordinates for a code with a generating matrix in the standard form is an information set. Nina Mostarac (nmavrovic@math.uniri.hr) 10/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples PD-sets Definition 5 Let C ⊆ F n p be a linear [ n , k , d ] code that can correct at most t errors, and let I be an information set for C . A subset S ⊆ Aut C is called a PD-set for C if every t -set of coordinate positions can be moved by at least one element of S out of the information set I . A lower bound on the size of a PD-set: Theorem 3.1 (The Gordon bound) If S is a PD-set for an [ n , k , d ] code C that can correct t errors, r = n − k, then: � n � n − 1 � � n − t + 1 � ��� | S | ≥ · · · · · · . r r − 1 r − t + 1 Nina Mostarac (nmavrovic@math.uniri.hr) 11/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Symmetric designs Definition 6 A symmetric ( v , k , λ ) -design is an incidence structure D = ( P , B , I ) which consists of the set of points P , the set of blocks B and an incidence relation I such that: • | P | = | B | = v , • every block is incident with exactly k points • and every pair of points is incident with exactly λ blocks ( λ > 0). A symmetric ( v , k , 1 ) -design is called a projective plane of order k − 1, and a symmetric ( v , k , 2 ) -design is called a biplane. Nina Mostarac (nmavrovic@math.uniri.hr) 12/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Incidence graph of a symmetric design Definition 7 An incidence graph or a Levi graph of a symmetric design is a graph whose vertices are points and blocks of the design, and edges are incident point-block pairs (flags). Remark 2 An incidence graph Γ of a symmetric ( v , k , λ ) -design: • is bipartite, • is k -regular, • has diameter diam (Γ) = 3. Nina Mostarac (nmavrovic@math.uniri.hr) 13/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Flag transitive symmetric designs Definition 8 • An automorphism of a symmetric design is a permutation of points which sends blocks to blocks. • An automorphism group of a symmetric design D is called flag-transitive if it is transitive on flags of D . Nina Mostarac (nmavrovic@math.uniri.hr) 14/20 PD-sets for codes related to flag-transitive symmetric designs

Introduction Codes from graphs Flag-transitive symmetric designs Examples Theorem 3.2 (Dankelmann, Key, Rodrigues [2](Result 7)) Let Γ = ( V , E ) be a k-regular graph with the automorphism group A transitive on edges and let G be an incidence matrix of Γ . If C = C p ( G ) is a [ | E | , | V | − ε, k ] p code, where p is a prime and ε ∈ { 0 , 1 , ... | V | − 1 } , then any transitive subgroup of A is a PD-set for full error correction for C. Theorem 3.3 (D.C., N.M.) Let Γ = ( V , E ) be an incidence graph of a symmetric ( v , k , λ ) -design D with flag-transitive automorphism group A and let G be an incidence matrix for Γ . Then C = C p ( G ) is a [ | E | , | V | − 1 , k ] p code, for any prime p, and any flag transitive subgroup of A can serve as a PD-set (for any information set) for full error correction for the code C. Nina Mostarac (nmavrovic@math.uniri.hr) 15/20 PD-sets for codes related to flag-transitive symmetric designs